The Graph Structure of Baker's maps Implemented on a Computer
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916424596848640 |
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| author | Li, Chengqing Tan, Kai |
| author_facet | Li, Chengqing Tan, Kai |
| contents | The complex dynamics of baker's map and its variants in an infinite-precision mathematical domain have been extensively analyzed in the past five decades. However, their real structure implemented in a finite-precision computer remains unclear. This paper gives an explicit formulation for the quantized baker's map and its extension into higher dimensions. Our study reveals certain properties, such as the in-degree distribution in the state-mapping network approaching a constant with increasing precision, and a consistent maximum in-degree across various levels of fixed-point arithmetic precision. We also observe a fractal pattern in baker's map functional graph as precision increases, characterized by fractal dimensions. We then thoroughly examine the structural nuances of functional graphs created by the higher-dimensional baker's map (HDBM) in both fixed-point and floating-point arithmetic domains. An interesting aspect of our study is the use of interval arithmetic to establish a relationship between the HDBM's functional graphs across these two computational domains. A particularly intriguing discovery is the emergence of a semi-fractal pattern within the functional graph of a specific baker's map variant, observed as the precision is incrementally increased. The insights gained from our research offer a foundational understanding for the dynamic analysis and application of baker's map and its variants in various domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04093 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Graph Structure of Baker's maps Implemented on a Computer Li, Chengqing Tan, Kai Chaotic Dynamics 94A55 The complex dynamics of baker's map and its variants in an infinite-precision mathematical domain have been extensively analyzed in the past five decades. However, their real structure implemented in a finite-precision computer remains unclear. This paper gives an explicit formulation for the quantized baker's map and its extension into higher dimensions. Our study reveals certain properties, such as the in-degree distribution in the state-mapping network approaching a constant with increasing precision, and a consistent maximum in-degree across various levels of fixed-point arithmetic precision. We also observe a fractal pattern in baker's map functional graph as precision increases, characterized by fractal dimensions. We then thoroughly examine the structural nuances of functional graphs created by the higher-dimensional baker's map (HDBM) in both fixed-point and floating-point arithmetic domains. An interesting aspect of our study is the use of interval arithmetic to establish a relationship between the HDBM's functional graphs across these two computational domains. A particularly intriguing discovery is the emergence of a semi-fractal pattern within the functional graph of a specific baker's map variant, observed as the precision is incrementally increased. The insights gained from our research offer a foundational understanding for the dynamic analysis and application of baker's map and its variants in various domains. |
| title | The Graph Structure of Baker's maps Implemented on a Computer |
| topic | Chaotic Dynamics 94A55 |
| url | https://arxiv.org/abs/2410.04093 |